Vectors.ppt

CHAPTER TWO:

VECTOR QUANTITIES

2. Vector Notation

3.Component of a vector 4. Properties of Vectors 5. Resultant Vector

In general

 _Vector

 _Magnitude



Boldface Letter



Letter with arrow above



(graphical) Straight line with arrowhead



VECTOR magnitude representation:



There are

3 ways

of writing vectors:

1.

AXIAL NOTATIONS

2.

NEWS NOTATION



write the magnitude and the angle it creates with



write the magnitude and

the appropriate direction

based on the NEWS

y

•_{rectangular coordinate} system (Axial notation)

•_{geographic reference frame} (NEWS)

x

N

W E

S

1 unit, 45 1 unit, NE

1 unit, 180 _{1 unit, west}

Vector

►

Equality of Two Vectors

►

Movement of vectors in a diagram

►

Any vector can be moved

parallel

to itself

without being affected

►

Negative Vectors

►_{is the vector sum of two or}

more vectors. It is the result of adding two or more vectors together. If displacement vectors A, B, and C are added together, the result will be vector R.

►

When adding vectors, their directions must

be taken into account



Graphical



Analytical

A. Graphical Method

1. Parallelogram Method

most applicable for two vectors

2. Polygon Method

applicable for three or more vectors

Parallelogram Method

1. Draw the two vectors.

2. Construct a parallelogram.

3. Construct a line from the origin that will bisect the parallelogram.

3. Construct a line.

A





B

2. Construct a parallelogram.

R

Polygon Method

1. Draw the vectors (head-to-tail). 2. Connect the tail of the first

vector to the head of the last vector.

The line that connect the tail of the first vector to the head of last

A





B

R

1. Draw the two vectors. (head-to-tail)

A

B

C

R

D

R = A + B + C + D

R = D + A + C + B

D

A

C

B. Analytical Method

(utilizes Mathematical concepts in analyzing vectors)

Component Method

►

A

component

is a

part

►

It is useful to use

►_{The x-component of a vector is the}

projection along the x-axis

►_{The y-component of a vector is the}

projection along the y-axis

►_Then,

cos

x

A



A



sin

y

A



A



x y

A

A





A

  

 = 60

A

A

Given the vector

(Vector A) in the

illustration:

►

The previous equations are valid

only if



is

measured with respect to the x-axis

►

The components can be positive or negative and will

have the same units as the original vector

►

The components are the legs of the right triangle

whose hypotenuse is

A

 _{May still have to find θ with respect to the positive x-axis}

x y 1 2 y 2 x A A tan and A A

QUADRANT

I

II

III

IV

X

+

-

-

+

-►_{Choose a coordinate system and sketch the vectors} ►_{Find the x- and y-components of all the vectors}

►_{Add all the x-components}

 _{This gives Rx:}





x

v

►_{Add all the y-components}

 _{This gives Ry:}

►_{Use the Pythagorean Theorem to find the magnitude of}

the Resultant:

►_{Use the inverse tangent function to find the direction of}

R:





v

R

R

R

R







simplifies the mathematics by

1.Express the vectors in terms of their component.

2. Add the components along the x direction to form the x

component of the resultant vector

3. Similarly, add the y components to get the y component of the resultant vector

If we have many vectors (A, B, C, D,...N), and we need to

find the resultant we just need to "SUM" all the x's and

all the y's separately, as illustrated

In the diagram, A has magnitude 12 units and

B has magnitude 8.0 units. The x component of A + B is about:

A

B

60º

45º

A= 3.0 units east

B= 4.0 units 30 N of W

C= 2.0 units 70 W of S

30

70

A

B

C

 _{A freshman student is lost in UPLB}

campus. From the PhySci building, his first two displacements are 1.0 km, 35° N of W and 500 m, 70° E of S, respectively. What is his third